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Let's consider an infinite percolation cluster $\mathcal{C}_p$ that takes shape in the supercritical phase ($p>p_c$) of a bond percolation in $\mathbb{Z}^d$. What could be said about a bond percolation on $\mathcal{C}_p$ parametrized by $p$ (or parametrized by a $q\ne p$)? Phase caracterizations? Critical values $q_c=f(p)$? Number of infinite clusters if any?

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    $\begingroup$ Can't you just imagine running percolation on $\mathbb{Z}^d$ twice (independently), and losing edges that get deleted either time? So you're running percolation with parameter $p^2$ (or $pq$)? $\endgroup$ Jun 1, 2012 at 16:53
  • $\begingroup$ I think you're right John! Nothing interesting is emerging from this percolation :-) I should have thought more carefully before posting this question! $\endgroup$
    – user16782
    Jun 1, 2012 at 19:41
  • $\begingroup$ It turns out, however, that carrying out percolation on the percolation clusters at criticality leads to an interesting model! arxiv.org/abs/1410.3603 $\endgroup$
    – j.c.
    Oct 15, 2014 at 1:01

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